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Doubling time

The doubling time is the time until a quantity has doubled. The rule of 70 says that you can divide 70 by the percentage and get an approximate number of periods. It is a rule of thumb, not an exact formula. The exact time is the logarithm of 2 divided by the logarithm of the growth factor. For 7% interest the rule gives 10 years, while the logarithm gives a little over 10 years. Use the logarithm when the task asks for an exact value.

t \approx \frac{70}{p}

the rule of 70

t = \frac{\ln 2}{\ln(1 + p/100)}

exact doubling time

Symbols

$p$	percent per period	%
$t$	number of periods

Example

7% interest. The rule of 70: $\dfrac{70}{7} = 10$ years.

Exact: $\dfrac{\ln 2}{\ln 1.07} \approx 10.24$ years.

10 years is the rule. 10.24 years is the logarithm. They are not the same calculation.

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Part of Mathematics 2P: Exponential growth.